= Three-point quadratic reproduction forces the identity filter
{title2=$(w_{-1},w_0,w_1)=(0,1,0)$}
A symmetric three-point smoother with weights $(a,b,a)$ reproduces constants only if $2a+b=1$. Reproducing quadratic trends also requires $2a=0$, so $a=0$ and $b=1$. Thus the identity <estimator> $X_t$ is unbiased, but there is no nontrivial three-point linear smoother reproducing every quadratic. An assertion excluding even the identity <estimator> is false. This clarifies why the <five-point symmetric quadratic trend smoother> is the first symmetric nontrivial example.
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